Is it possible to move VMs from let's say ESXi 6.7 to ESXi 7.0.3 host and have them run on the new host without issues?
The VMs are stored in local storage if that makes a difference.
Yes, of course you can. Use a backup solution called Vinchin Backup & Recovery that supports ESXi 5.0 ~ 8.0, you can back up ESXi 6.7 and restore the backups to ESXi 7.0.3, then you will get the new VM populated with the data. And you don't need any command utility or preconditions like compatibility or environment preparation. Take a try! I've tried it and it's great.